A proof of Powell's conjecture on the Goeritz group of $S^3$

Daiki Iguchi

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arXiv · 2605.21905

A proof of Powell's conjecture on the Goeritz group of $S^3$

Daiki Iguchi

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Abstract

For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$.

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Daiki Iguchi. 2026-05-21. A proof of Powell's conjecture on the Goeritz group of $S^3$. https://arxiv.org/abs/2605.21905

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